On the Formal Comparison of Milne's Kinematical System with the Systems of General Relativity

Abstract

I 935 J an -Milne's Kinematical System and General Relativity 263 function H. An alternative way of proceeding would be to avoid development in terms of functions u n and to have recourse to the original equation (I). Guided by the results of the preceding sections, the solution u, as regards its main part, might be put into the form u=(f> cos (Xt + e) + x cos 2 1. When Milne introduced his kinematical system for the description of the gravitational field of the universe, the question at once became im-portant whether this model differed essentially from the variety of models already provided by general relativistic cosmology. The differences that may exist between such models can be investigated by two different methods. The first method is to find the relations between the various astronomical observables in one system and to compare these with the corresponding relations in the other systems.* The importance of this method is that it provides possible tests of the different theories, but apart from this, it is unsatisfactory owing to the restricted number of such observables. The second method is to make & formal comparison of the systems, and this is the object of the present paper. We shall construct in accordance with the general theory of relativity a material system which is identical with Milne's system, and then formally compare the gravitational fields of this system predicted by the two theories. That these fields differ was the conclusion of a previous paper f by the author. In that paper, Milne's equations of motion for a free particle in the presence of a certain distribution of matter in motion were expressed in the form of a principle of least action. The result at once indicated that the geometry appropriate for a representation of Milne's system is Finsler and not Riemannian geometry. 2. The first step is to find the metric which, according to the general theory of relativity, corresponds to the material system studied by Milne. This system consists of a set of fundamental particles attached to each of which is an observer, and these particles satisfy the cosmological principle, i.e. each fundamental observer sees the same sequence of world-pictures. From temporal experiences only, the observers provide themselves with clocks which may be said to be similar and synchronized. Each observer then determines the epoch and distance of a distant event from observations made on his own clock according to the following conventions. If a light signal sent by an observer A at time ^ by his clock is reflected at an event E and received by A at time t 2 , then A assigns to E an epoch T and distance X where T = J(í 2 + íi) X = \c{t 2 -t-¡). (1)